The curvature of convex sum of metrics and applications
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arXiv
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| Format: | Preprint |
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2021
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| _version_ | 1866916025991168000 |
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| author | Cavenaghi, Leonardo F. Galindo, Giovane Sperança, Llohann D. |
| author_facet | Cavenaghi, Leonardo F. Galindo, Giovane Sperança, Llohann D. |
| contents | In this note, we derive explicit formulae for the curvature of a convex sum of Riemannian metrics, \(g_t = (1-t)g_0 + t g_1\). We study whether such a deformation can increase the \emph{average} of the Riemann curvature component \(R_t(X,Y,Y,X)\) along an immersed, totally geodesic flat torus. Because a first-order increase is prohibited, we obtain necessary and sufficient conditions for \(g_t\) to have a positive average variation of order \(r \geq 2\). These conditions are applied to paths joining \(g_0\) to classical metric deformations, including conformal changes, vertical warpings, and Cheeger deformations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2106_14781 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | The curvature of convex sum of metrics and applications Cavenaghi, Leonardo F. Galindo, Giovane Sperança, Llohann D. Differential Geometry In this note, we derive explicit formulae for the curvature of a convex sum of Riemannian metrics, \(g_t = (1-t)g_0 + t g_1\). We study whether such a deformation can increase the \emph{average} of the Riemann curvature component \(R_t(X,Y,Y,X)\) along an immersed, totally geodesic flat torus. Because a first-order increase is prohibited, we obtain necessary and sufficient conditions for \(g_t\) to have a positive average variation of order \(r \geq 2\). These conditions are applied to paths joining \(g_0\) to classical metric deformations, including conformal changes, vertical warpings, and Cheeger deformations. |
| title | The curvature of convex sum of metrics and applications |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2106.14781 |